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arXiv · 2203.06238

When does the Auslander-Reiten translation operate linearly on the Grothendieck group? -- Part I

Carlo Klapproth
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Abstract

For a hereditary, finite-dimensional algebra $A$ the Coxeter transformation extends the action of the Auslander--Reiten translation on the non-projective indecomposable modules to a linear endomorphism of the Grothendieck group of the category of finitely generated $A$-modules. It is natural to ask whether other algebras admit a similar linear extension. We show that this is indeed the case for all Nakayama algebras. Conversely, we show that finite-dimensional algebras with non-acyclic and connected quiver admitting such a linear extension are already cyclic Nakayama algebras.

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Carlo Klapproth. 2023-09-01. When does the Auslander-Reiten translation operate linearly on the Grothendieck group? -- Part I. https://doi.org/10.1016/j.jpaa.2023.107571

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